A model category has two weak factorizations, a pair of cofibrations and trivial fibrations and a pair of trivial cofibrations and fibrations. Then the class of weak equivalences is the set W consisting of the morphisms that can be decomposed into trivial cofibrations followed by trivial fibrations. One can build a model category out of such two weak factorizations by defining the class of weak equivalences by W as long as it satisfies the two out of three property. In this note we show that given a category with two weak factorizations, if every object is fibrant and cofibrant, W satisfies the two out of three property if and only if W is closed under the homotopies.
Keywords: Model category, Quillen category, weak equivalence, two out of three property, homotopy
2020 MSC: Primary 18N40; Secondary 55U35
Theory and Applications of Categories, Vol. 38, 2022, No. 1, pp 1-26.